Polybaric fractional crystallisation of arc magmas: an experimental study simulating trans-crustal magmatic systems

Crystallisation-driven differentiation is one fundamental mechanism proposed to control the compositional evolution of magmas. In this experimental study, we simulated polybaric fractional crystallisation of mantle-derived arc magmas. Various pressure–temperature trajectories were explored to cover a range of potential magma ascent paths and to investigate the role of decompression on phase equilibria and liquid lines of descent (LLD). Fractional crystallisation was approached in a step-wise manner by repetitively synthesising new starting materials chemically corresponding to liquids formed in previous runs. Experiments were performed at temperatures ranging from 1140 to 870 °C with 30 °C steps, and pressure was varied between 0.8 and 0.2 GPa with 0.2 GPa steps. For most fractionation paths, oxygen fugacity (fO2) was buffered close to the Ni-NiO equilibrium (NNO). An additional fractionation series was conducted at fO2 corresponding to the Re-ReO2 buffer (RRO ≈ NNO+2). High-pressure experiments (0.4–0.8 GPa) were run in piston cylinder apparatus while 0.2 GPa runs were conducted in externally heated pressure vessels. Resulting liquid lines of descent follow calc-alkaline differentiation trends where the onset of pronounced silica enrichment coincides with the saturation of amphibole and/or Fe–Ti–oxide. Both pressure and fO2 exert crucial control on the stability fields of olivine, pyroxene, amphibole, plagioclase, and Fe–Ti–oxide phases and on the differentiation behaviour of arc magmas. Key observations are a shift of the olivine–clinopyroxene cotectic towards more clinopyroxene-rich liquid composition, an expansion of the plagioclase stability field and a decrease of amphibole stability with decreasing pressure. Decompression-dominated ascent trajectories result in liquid lines of descent approaching the metaluminous compositional range observed for typical arc volcanic rocks, while differentiation trends obtained for cooling-dominated trajectories evolve to peraluminous compositions, similar to isobaric liquid lines of descent at elevated pressures. Experiments buffered at RRO provide a closer match with natural calc-alkaline differentiation trends compared to fO2 conditions close to NNO. We conclude that decompression-dominated fractionation at oxidising conditions represents one possible scenario for arc magma differentiation.

During the last decades, deep and shallow differentiation of parental mafic magmas via crystallisation has been investigated experimentally. A crucial observation of experimental high-pressure liquid lines of descent (0.7-1.2 GPa) established for fractional Ulmer et al. 2018) or equilibrium crystallisation (Müntener et al. 2001;Alonso-Perez et al. 2009;Blatter et al. 2013) is the rapid increase of the alumina saturation index ASI (= Al 2 O 3 / (CaO + Na 2 O + K 2 O)) with ongoing differentiation resulting in peraluminous (ASI > 1) intermediate (andesitic) melt compositions. This general behaviour is the result of extensive clinopyroxene and amphibole crystallisation at high pressures depleting residual liquids efficiently in calcium but to a much lesser extent in alumina. However, the majority of intermediate volcanic and plutonic arc rocks are metaluminous and only highly evolved silica-rich lithologies become weakly peraluminous. On the contrary, liquid lines of descent defined at low pressure (0.2-0.4 GPa) via equilibrium crystallisation Grove 1993a, 1993b;Grove et al. 2003;Blatter et al. 2013) provide a closer match with the compositional trend exhibited by natural rocks and remain metaluminous upon progressive differentiation.
To resolve the open question of the predominant pressure level or crustal depth of crystallisation differentiation, we experimentally investigated the process of polybaric fractional crystallisation, where hydrous basaltic liquids extracted from the mantle differentiate at various levels in the crust (e.g. Grove et al. 2003;Almeev et al. 2013;Melekhova et al. 2015). Primary mantle-derived basaltic magmas buoyantly rise through the upper mantle and initiate differentiation via fractional crystallisation close to the crust-mantle boundary. Derivative evolved basaltic or andesitic liquids subsequently rise to the middle and upper crust and simultaneously continue to fractionate. Here we envisage that crystallisation differentiation occurs over a range of crustal pressures and temperatures generating different residual melt compositions. Since Clapeyron slopes of mineral reactions in calc-alkaline systems are rather different from near-adiabatic ascent trajectories (e.g. Blundy and Cashman 2001), different mineral boundaries are crossed during decompression. Most prominent and crucial for this study is a destabilisation of clinopyroxene at the expense of olivine with decreasing pressure via a displacement of the olivine-clinopyroxene cotectic curve towards melt compositions with higher normative clinopyroxene contents (e.g. Grove and Baker 1984;Grove et al. 1992;Stamper et al. 2014). In addition, decreasing pressure expands the stability field of plagioclase with respect to clinopyroxene and olivine preventing a pronounced enrichment of residual melts in alumina (e.g. Sisson and Grove 1993a;Grove et al. 2003;Alonso-Perez et al. 2009). Water solubility in silicate liquids diminishes with decreasing pressure (e.g. Hamilton et al. 1964) inferring that water activities (aH 2 O) continuously increase for H 2 O-undersaturated hydrous arc magmas during decompression. Such a change in aH 2 O has an additional effect on plagioclase composition favouring the crystallisation of anorthite-rich plagioclase (e.g. Sisson and Grove 1993a). As a consequence of the combined effect of pressure-sensitive changes on phase equilibria, polybaric fractional crystallisation is expected to prevent calc-alkaline liquids to evolve towards peraluminous compositions at low SiO 2 , closer approaching the natural rock record.
Here we report the results of a set of polybaric fractional crystallisation experiments conducted on hydrous calc-alkaline basalts by varying pressure and temperature concomitantly along the liquid line of descent (LLD). We explored different polybaric fractionation ascent trajectories and conducted a limited number of additional experiments to evaluate the effect of oxygen fugacity (fO 2 ) and initial starting material composition on the resultant liquid lines of descent. We discuss the effect of polybaric fractionation on calc-alkaline magmas and propose a general petrological model for arc magma differentiation.

General strategy
Fractional crystallisation (FC) was simulated in a stepwise manner following established techniques (Villiger et al. 2004;Nandedkar et al. 2014;Ulmer et al. 2018): for each fractionation step, the chemical composition of the residual liquid (quenched to a homogeneous glass) of the respective experiment is determined by EPMA and a new, chemically corresponding starting material is synthesised employing pure chemicals. This starting material is then used for the subsequent experiment at different pressure-temperature conditions. Experiments were run between 1140 and 870 °C, at pressures ranging from 1.0 to 0.2 GPa. Temperature steps of 30 °C (in a few cases 15 °C) and pressure intervals of 0.2 GPa were employed. The polybaric approach required that pressure and temperature had to be varied simultaneously. However, P-T ascent trajectories of rising magmas through the crust are poorly constrained (e.g. Rutherford 2008;Neave and Maclennan 2020) and, therefore, three main experimental series with different P-T trajectories have been investigated: (1) a hot path where each fractionation was linked to a decompression step simulating an ascent trajectory of 22.5 °C/0.1 GPa; (2) an intermediate path with one additional fractionation step performed under isobaric conditions (30 °C/0.1 GPa), and (3) a cold path where two isobaric fractionation steps were conducted between each decompression step (45 °C/0.1 GPa). An additional fractionation series (hot rk52 path) followed a similar P-T ascent trajectory as the hot path but initiated from a different starting material to test whether the normative clinopyroxene content of the primary starting material exerts a crucial impact on the resultant liquid line of descent during polybaric fractionation. Finally, a hot oxidised path (with an ascent trajectory of 19 °C/0.1 GPa) was explored, where fO 2 was buffered close to the Re-ReO 2 equilibrium (RRO), while fO 2 was constrained close to the Ni-NiO buffer (NNO) for all other series. The P-T ascent trajectories given above were calculated by fitting a line through the first and last experiment of each fractionation series. Assuming an average crustal density of 2800 kg/m 3 , the hot trajectories (hot, hot rk52, and hot oxidised path) simulate cooling rates of approx. 6-7 °C/km during magma ascent, the intermediate path corresponds to 8 °C/km and the cold path simulates 12 °C/km. Explored P-T trajectories are illustrated in Fig. 1, where fractionation steps are highlighted by an arrow and equilibrium steps are indicated by simple lines. Intermediate equilibrium crystallisation steps were required when less than 10 wt.% of minerals crystallised in the fractionation experiments to promote more extensive crystallisation prior to the execution of the next fractionation step.

Starting materials
Derivative liquids from the fractional crystallisation study at 1.0 GPa by Ulmer et al. (2018) on a primitive high-Mg basalt from the Adamello Batholith (Northern Italy) were used as initial starting materials. Four fractionation series (hot, hot oxidised, intermediate, and cold path) were run on the residual liquid of the run rk54 (1140 °C) exhibiting an xMg of 0.585, where xMg equals the molar ratio of MgO/ (FeO tot + MgO), with all iron assumed to be ferrous. The hot rk52 path was performed on the melt formed in experiment rk52 (1170 °C, xMg of 0.638). Both compositions were derived from a primary high-Mg basalt (RC158c) via fractionation of the mineral phase assemblage olivine, clinopyroxene, and spinel. We slightly modified the liquid compositions reported by Ulmer et al. (2018) by adding P 2 O 5 as a chemical component (0.16 wt.% for rk54 and 0.13 wt.% for rk52) and decreasing H 2 O contents to 3.50 wt.% (rk54) and 2.96 wt.% (rk52), respectively. For each fractionation step, FeO and Na 2 O (and in some cases also K 2 O and P 2 O 5 ) contents of new starting materials were re-calculated from compositional data of experimental liquids and mineral phases employing phase proportions established by mass balance calculations to account for potential Fe-loss and alkali migration during EPMA analysis. Fe 2+ /Fe 3+ ratios of starting materials were adjusted to values corresponding to the NNO or RRO oxygen buffers using the algorithm by Kress and Carmichael (1991). Water contents of starting materials were calculated by mass balance treating water as a perfectly incompatible compound in nominally anhydrous minerals and assuming an empirical H 2 O content of 2 wt.% for amphibole. For low-pressure experiments (0.2 and 0.4 GPa), water contents of new starting materials were compared with expected water contents at saturation for respective experimental P-T conditions calculated by MagmaSat (Ghiorso and Gualda 2015) to avoid significant amounts (several wt.%) of excess free water in the experimental charges influencing the silicate-fluid partitioning behaviour of some lithophile elements (e.g. Na and K). However, for the last starting material of the intermediate path

Capsule design and capsule preparation
A double capsule technique representing a modification and combination of the designs by Kaegi et al. (2005) and Matjuschkin et al. (2015) was employed, where two inner capsules (outer diameter, OD of 2.3 mm) were placed in an outer capsule (OD of 4.0 mm). The first inner capsule was prepared from Au 90 Pd 10 or Au 100 (below 1000 °C) and contained the experimental charge.

Experimental procedure
Experiments at 1.0, 0.8, and 0.6 GPa were run in an endloaded 14-mm bore Boyd and England-type piston cylinder apparatus at ETH Zürich. NaCl-Pyrex-graphite-MgO assemblies were utilised, and a friction correction of 5% was applied to the nominal pressure. Capsules were placed in the hotspot of the assembly, whose position was estimated using the numerical algorithm of Hernlund et al. (2006). Temperature was measured with a B-type thermocouple (Pt 94 Rh 6 / Pt 70 Rh 30 ) with an estimated accuracy of ± 10 °C. No correction for the pressure effect on the electromotive force (e.m.f.) was applied. Numerical modelling revealed that hotspot temperatures were 7-16 °C higher than the effective thermocouple reading. Oil pressure was automatically controlled with an in-house built worm-gear-type pressure control system with a tolerance window of 0.5 bar oil pressure (≈ 40 MPa). After capsule loading, assemblies were initially pressurised to approximately 50 bar oil pressure (≈ 0.4 GPa) at ambient temperature and then heated with a continuous rate of 40 °C/ min to final run temperatures. Pressure was slowly increased when temperature exceeded 500 °C (softening point of Pyrex glass), but care was taken to reach final pressure before the attainment of the run temperature. Experimental run times varied from 6 to 73 h with longer durations at lower temperatures. Finally, experiments were quenched by switching off the power supply resulting in cooling rates exceeding 100 °C/s. Experiments at 0.4 GPa were performed in a 22-mm bore non-end-loaded Johannes-type piston cylinder at ETH Zürich, which was specifically calibrated for low-pressure conditions employing the NaCl melting curve (Clark 1959) in combination with the falling sphere technique (e.g. Baker 2004). Run protocol and control of pressure and temperature were identical to experiments run at higher pressures in the 14-mm bore Boyd and England-type piston cylinder, with the exception that a slightly different NaCl-Pyrex-graphite-MgO assembly was utilised (see Electronic Supplementary Material ESM 2). Thereby, capsules were additionally embedded in Pyrex glass powder to ensure homogenous pressure distribution. After loading, experiments were pressurised to approximately 0.15 GPa, and subsequent heating procedure was identical to the end-loaded piston cylinder approach. Cooling rates during quenching were slightly lower (50-100 °C/s) compared to the 14-mm assembly used for the experiments at higher pressures due to the larger volume of the 22-mm assembly. Experiments at 0.2 GPa were run in externally heated Molybdenum-Hafnium-Carbide (MHC) pressure vessels at ETH Zürich. A gas mixture of Ar-CH 4 was used as pressurising medium, where methane acted as a reactant component buffering hydrogen fugacity and, thus, fO 2 in the run charges. Methane contents of the gas mixture were adjusted to maintain fO 2 conditions close to the NNO or RRO buffer. General run procedure was as follows: after capsule loading, vessels were first filled with appropriate amounts of methane and subsequently pressurised with argon to levels corresponding to roughly half of the final run pressure. Then heating was started, and gas pressure increased simultaneously with temperature due to thermal expansion. After run temperature was reached, pressure was adjusted manually to the final target level of 200 MPa. During experiments, pressure was monitored using a pressure sensor with an uncertainty of 0.5% and manually controlled at 200 MPa with a maximum deviation of ± 2 MPa. Temperature was controlled with a K-type thermocouple, which was positioned outside of the MHC vessel above the approximate capsule position. Temperature recorded with this external thermocouple was calibrated against effective experimental temperature inside the bomb every time a new vessel was installed. Generally, internal and external temperature readings exhibit a linear relationship, and 2σ errors on established temperature calibrations were below 10 °C. During experiments, vessels were oriented with an inclination of 10° to ensure the development of a stable temperature profile. All experiments run in the MHC pressure vessels were equilibrated at run conditions for 72 h. Quenching was attained by tilting the furnace by 90° into a vertical position forcing the capsule to fall to the cold end of the vessel, resulting in quenching rates exceeding 100 °C/s. For one experiment (FM146), temperature cycling (e.g. Mills and Glazner 2013) was performed to promote the formation of melt pools large enough for reliable analysis. Thereby, temperature was cycled with ± 10 °C around the target equilibrium temperature (960 °C) for half of the intended run duration. Employed heating and cooling rates during temperature cycling were 0.5 °C/min and temperature plateaus were hold for 40 min. After this initial cycling phase, temperature was adjusted to 960 °C and kept for the remaining time of the experiment.

Sample preparation and analytical techniques
Recovered capsules were embedded in epoxy-resin, opened with grinding paper to expose a longitudinal cross section, and subsequently polished using diamond suspensions with finest grades of 1 μm. Polished charges were inspected under an optical microscope for capsule coherence and any peculiar textures. In addition, buffer capsules were checked for the presence of metal and oxide phases to verify the fO 2 buffering capability of the buffer assemblage mixture 3 Page 6 of 36 throughout the entire run duration. Subsequently, specimens were coated with 20 nm of carbon and analysed with a JEOL JSM-6390 Scanning Electron Microscope at ETH Zürich. Runs were checked for general appearance, and high-quality back-scatter electron (BSE) images were acquired. First quantitative compositional data of glasses and minerals were obtained with a standardised Thermo Fisher NORAN NSS7 Energy-Dispersive Spectroscopic system with a 30 mm 2 Silicon-drift detector (SDD) (SEM-EDS). Acceleration voltage for imaging and measurements was set to 15 kV. In a final step, major phases (glass, olivine, hercynitic spinel, plagioclase, ortho-and clinopyroxene, amphibole, biotite, magnetite, ilmenite, and apatite) were analysed with two JEOL Electron Probe Micro Analysers (EPMA) at ETH Zürich. Approximately half of the experiments presented in this project were measured with a JEOL JXA 8200 Superprobe, while a JEOL JXA 8230 Superprobe was used for the second half. The same set of oxide and silicate standards was utilised for both devices, and a comparison of analytical results revealed no significant differences. An acceleration voltage of 15 kV was utilised for analysis. Sodium and potassium were measured first to avoid extensive alkali migration. Hydrous glasses were analysed with a beam current of 7 nA and a beam diameter of 20 μm. In case residual melt pools were too small, beam currents were reduced to 4 or 2 nA and spot sizes to 10 or 5 μm. Plagioclase, biotite, and apatite were analysed with a focussed beam and a current of 10 nA, while the other phases (pyroxenes, amphibole, olivine, spinel, magnetite, and ilmenite) were measured with a focussed beam and a current of 20 nA. For fO 2 reconstruction, iron contents of AuPd capsule material pieces were measured with a beam current of 20 nA and pure metal standards for Au and Pd and wustite for Fe.

Results
An overview of experimental run conditions is given in Table 2, including employed starting materials, capsule materials, stable phase assemblages, phase proportions, relative Fe-losses, estimated melt water contents, and calculated minimum fO 2 conditions based on water activity. These data are also supplied in an electronic form in the Electronic Supplementary Material ESM 3.

General remarks
All successful experiments consisted of residual liquid quenched to a homogenous glass and a variety of mineral phases, namely, olivine (ol), clinopyroxene (cpx), orthopyroxene (opx), plagioclase (plag), hercynitic spinel (sp), amphibole (amph), magnetite (mt), biotite (bt), ilmenite (ilm), and apatite (ap). Although most experiments were nominally H 2 O-undersaturated, fluid bubbles were observed in all runs, indicating that fluid-saturated conditions were met. Pure H 2 O-saturation was reached by a few experiments performed at low temperatures (< 1000 °C) and low pressures (0.2 or 0.4 GPa, compare Table 2). The occurrence of a free fluid phase in the H 2 O-undersaturated experiments is most probably the consequence of the presence of additional fluid components in the charges with low solubilities in silicate liquids. One likely candidate is CO 2 which was introduced via absorption by the fine-grained synthetic starting material powders . Although starting materials were stored at 110 °C before capsule loading and attention was paid to process new starting materials quickly after synthesis, the presence of CO 2 in the final charge could not be avoided. A further likely contaminant fluid component is N 2 that was incorporated from the surrounding air during capsule preparation. However, results from a crosscheck experiment on a pre-vitrified starting material (± free of CO 2 and N 2 ) infer that the presence of small amounts of these additional volatile compounds had no discernible impact on phase equilibria and compositions of residual melts and coexisting minerals (Marxer 2021).
A prerequisite for successful experimentation is the maintenance of a chemically closed system. Two compounds that potentially escape and/or contaminate the experimental charge at run conditions are iron and hydrogen. We quantified iron loss for each run via mass balance regression, treating all iron alloying with the capsule as an additional phase for calculations. In addition, for runs containing an Fe-Ti-oxide phase (complicating mass balance calculations), proportions of ilmenite or magnetite were independently determined via image processing on BSE images of entire charges employing the software ImageJ. First, the capsule material as well all voids and cracks were manually removed via blackening. Subsequently, oxide proportions were estimated assuming that area fractions equal volume fractions. Finally, volume proportions were converted to mass fractions employing calculated densities of all coexisting phases. Densities of residual melts were established using the algorithm of Lange and Carmichael (1987), while densities of mineral phases were derived employing a Birch-Murnaghan equation of state and an own compilation of required thermodynamic parameters. Fe-losses were finally recalculated via mass balance employing oxide fractions obtained by image processing. For two experiments (FM146 and FM121), established iron losses were negative indicative of iron gain, illustrating the general drawback of image analysis procedures as the underlying assumption of homogeneous distribution of oxide phases in the experimental charge is most probably not entirely correct. Magnetite tends to preferentially nucleate on capsule walls, and previous studies reported accumulation of magnetite crystals on volatile bubbles by floating (e.g. Knipping et al. 2019). melt, plag, ol, mt, amph(t), ilm(t) 88.7(13); 6.6(10); 3.9(6); Accounting for this uncertainty by assuming a relative error of 20% on Fe-Ti-oxide fractions results in maximum estimated Fe-loss close to zero. Consequently, we assumed that Fe-loss was negligible for these runs and report Fe-Ti-oxide proportions estimated via mass balance (Table 2). This is consistent with the relatively low temperatures of these runs (< 1000 °C) and the use of pure Au capsules, representing well-known experimental conditions to minimise iron loss (e.g. Barr and Grove 2010). Generally, established Fe-losses were below 15.5% (relative to the FeO content of the employed starting material) for all fractionation experiments. However, especially for runs performed at high temperatures (≥ 1020 °C) in Au 90 Pd 10 capsules, loss of iron was significant due to an increasing maximum solubility of iron in AuPd alloys with increasing temperature and palladium content (Barr and Grove 2010). Furthermore, noble metal capsules are permeable for hydrogen under experimental P-T conditions leading to either H-gain or H-loss via diffusion during run conduction (e.g. Eugster 1957;Eugster and Wones 1962). The solid buffer assemblage (NNO or RRO) imposes an intrinsic hydrogen fugacity (fH 2 ) which equilibrates throughout the entire outer capsule thereby constraining a specific fO 2 in the second inner capsule containing the experimental charge. Consequently, hydrogen loss is expected to be negligible for this study as long as the solid-state buffer was capable of maintaining a roughly constant fH 2 throughout the entire outer capsule. Generally, Fe-loss to the inner capsule results in the formation of free oxygen which either oxidises ferrous iron to ferric iron or reacts with the hydrogen supplied from the fO 2 buffer to H 2 O. Consequently, pronounced iron loss can lead to an increase of water contents of experimental charges as long as fH 2 is buffered. However, we infer that due to the elevated water contents of the employed starting materials of this study (> 3.5 wt.%) the amount of additional H 2 O formed is generally negligible, since the highest estimated iron loss (15.5%) could only have supplied a maximum of 0.35 wt.% of additional water.
Water contents of residual melts were quantified for few selected runs via Raman spectroscopy. Analytical procedures and results are presented in the Electronic Supplementary Material ESM 2. However, consistent phase assemblages and chemical trends of experimental runs indicate that H 2 O contents evolve systematically along the LLD as inferred by mass balance considerations. This is supported by a continuous increase of glass water contents estimated via the EPMA "by-difference" method (e.g. Anderson 1973;Blundy and Cashman 2008;Hughes et al. 2019) with decreasing temperature. However, H 2 O contents established via the "by-difference" approach are approximately 2 wt.% higher compared to mass balance calculations, which we assign to a matrix effect during EPMA analysis due to the standardisation on mineral phases. For experiments containing plagioclase we utilised the Waters and Lange (2015) plagioclase-melt hygrometer to calculate melt water contents that match mass balance results at concentrations below 6.0 wt.% H 2 O to within 10%. At higher nominal H 2 O contents, this disparity increases to 1.0-3.0 wt.% pointing towards limitations of the Waters and Lange (2015) algorithm at elevated melt water contents (and pressures).

Oxygen fugacity
Most runs were H 2 O-undersaturated and, thus, the fO 2 conditions in the run capsules were displaced to lower values compared to the nominal buffer equilibria (NNO or RRO) by the factor 2*log(aH 2 O) (where aH 2 O corresponds to the activity of water in the experimental charge). Only, when experimental runs reached water-saturation (aH 2 O = 1), which was the case for some low-temperature and lowpressure runs, buffered redox conditions corresponded to NNO or RRO. Consequently, we estimated minimum fO 2 of our experiments employing the Burnham water solubility and activity model (e.g. Burnham and Davis 1974;Burnham 1994) following the procedure described in Holloway and Blank (1994). For each run, water contents of recovered residual glasses were calculated via mass balance using mineral phase proportions and H 2 O concentrations of employed starting materials. Water activities were subsequently calculated employing the Burnham algorithm. Melt compositional parameters and derived water activities are summarised in the Electronic Supplementary Material ESM 4. However, the Burnham algorithm tends to underestimate H 2 O solubility at higher pressures (e.g. Holtz et al. 1995;Botcharnikov et al. 2005) resulting in a slight overestimation of aH 2 O. The water activities were then used to estimate experimental redox conditions. Calculated fO 2 values are reported relative to NNO (Table 2). They represent minimum conditions since failure of the inner buffer capsule would result in a slight increase in fO 2 .
The attainment of targeted experimental redox conditions was verified with several different, independent approaches. Figure 2 summarises fO 2 conditions expressed relative to the NNO equilibrium. For a limited number of experiments, fO 2 conditions were reconstructed via the solubility of Fe in AuPd alloys employing the solution model of Barr and Grove (2010). For this purpose, iron contents of small noble metal pieces (Au 90 Pd 10 ) dispersed in the experimental charges were measured by EPMA. For some runs, a rather large variability of alloy compositions (and resultant redox conditions) was observed. Therefore, we only report ranges of reconstructed fO 2 . The lowest fO 2 corresponds to the highest measured iron content, and vice versa. The large spread of AuPdFe compositions is mainly related to the uncertainty on the attainment of equilibrium between silicate melt and noble capsule 3 Page 10 of 36 metal. Consequently, we infer that the lower ends of the fO 2 range represent the most realistic values for experimental fO 2 conditions. In general, fO 2 conditions vary between NNO-1 and NNO+1 with a majority plotting in a narrower range of NNO±0.5. When only considering minimum values, the range of fO 2 conditions narrows to NNO-1 to NNO, in agreement with minimum redox conditions of experimental charges inferred from water activities and redox buffers (Table 2). Measured alloy compositions, calculated fO 2 conditions, and estimated errors are reported in the Electronic Supplementary Material ESM 5. In addition, experimental fO 2 conditions were estimated for amphibole-bearing charges employing the thermobarometric algorithms of Ridolfi et al. (2010) (R10) and Ridolfi and Renzulli (2012) (RR12). The R10 model results in consistently lower fO 2 than the RR12 model. Results from the R10 formulation vary between NNO-0.1 and NNO+1.1, while values calculated with RR12 range from NNO+1.0 to NNO+2.4. No consistent trends of calculated fO 2 with experimental temperature or pressure were observed and, thus, only ranges of established fO 2 are reported (Fig. 2). In agreement with Erdmann et al. (2014), we infer that the RR12 algorithm tends to overestimate fO 2 conditions and conclude that redox conditions determined with the R10 model are more realistic. Finally, in one experiment (FM116, 960 °C and 0.6 GPa), magnetite and ilmenite coexist and, thus, Fe-Ti-oxide oxybarometry was applied. The algorithm of Ghiorso and Evans (2008) resulted an Fe-Ti exchange temperature of 954 °C and a logfO 2 of NNO-0.25. Both values agree within error with experimental conditions, confirming that the experiments were buffered at redox conditions close to NNO.

Run appearance
The step-wise experimental approach to fractional crystallisation generated high residual melt fractions exceeding 40-50 wt.% (in most cases > 70 wt.%) facilitating the analysis of hydrous glasses. For most experiments, mineral phases were homogeneously distributed over exposed capsule sections. In few cases, weak gradients in phase abundances (e.g. zones with higher melt fractions) were observed and attributed to small temperature gradients (< 10 °C over the length of entire outer capsules). Additionally, glasses and mineral phases were analysed throughout, and no systematic compositional variations were observed. The employment of different experimental techniques and apparatus for the polybaric fractionation approach resulted in a distinct textural variability of recovered charges, which is described in detail in the following paragraphs and illustrated in Fig. 3.
In general, the formation of quench phases (e.g. plagioclase, clinopyroxene, amphibole, or biotite) was of minor extent in runs performed in the 14-mm Boyd and Englandtype piston cylinder (0.8 and 0.6 GPa), slightly more extensive in the experiments run in the 22-mm non-end-loaded Johannes-type piston cylinder (due to the slower quenching rates), and completely absent for experiments conducted at 0.2 GPa in the MHC pressure vessels. Quench crystallisation was either observed as overgrowth rims on stable mineral phases (e.g. high-albite plagioclase rims or clinopyroxene/amphibole overgrowth on equilibrium clinopyroxene) or direct nucleation of new crystals from the groundmass glass (biotite laths). Relics of unreacted starting material are almost entirely absent except in run FM164 where Al 2 O 3 relics were identified as small inclusions in plagioclase clusters. Olivine was equant and often exhibited rounded shapes ( Fig. 3a, b and e), which we interpret as a result of crystallisation kinetics of olivine in magmatic experiments, where olivine generally tends to crystallise extensively during the initial phase of an experiment, while the nucleation and growth of other phases are retarded. As soon as coexisting stable mineral phases start crystallising, olivine composition and modal abundance re-adapt to changing melt composition via partial resorption to approach final equilibrium. Clinopyroxene was mostly small and equant (< 10 µm) and sometimes exhibited bright cores with low Al contents, which we interpret as metastable cores formed during nucleation (Fig. 3a). Therefore, care was taken to avoid such cores during EPMA analysis. Extending run duration resulted in successful reduction of the abundance of these low-Al cores. In two experiments (FM93 and FM146) sector zoned clinopyroxenes (Fig. 3f), were identified, a feature that has previously been reported in experimental studies (e.g. Sisson and Grove 1993a; Ulmer et al. 2018) and natural rocks (e.g. Hurlimann et al. 2016). We infer that the formation of sector zoning is a result of rapid crystal growth close to the upper stability limit of clinopyroxene favouring crystallisation over nucleation (Schwandt and Mckay 2006). For run FM146, clinopyroxene sector zoning was likely promoted by temperature cycling during the experiment. Plagioclase generally formed small subhedral equant or lath-shaped crystals (< 10 µm). Zoning of single crystals could not be observed. At 0.2 GPa, plagioclase was distinctively smaller (< 5 µm) than at higher pressures, resulting in finely dispersed plagioclase in the groundmass glass (Fig. 3e, f). One reason for these specific textures is the pronounced oversaturation in plagioclase at low pressures resulting in rapid nucleation when equilibration P-T conditions are reached. Hercynitic spinel formed small, euhedral crystals (< 5 µm), which were sometimes difficult to identify in BSE images due to greyscale levels overlapping with clinopyroxene. Amphibole was idiomorphic (Fig. 3c, d), and a decrease of grain sizes with decreasing temperature (from around 100 to 10 µm) was observed. This variation in crystal size can be explained by a change of the nucleation vs crystallisation rates. At high temperatures, close to its stability limit, growth of amphibole is favoured, while at lower temperatures nucleation dominates. This interpretation is supported by the presence of melt and mineral inclusions in amphibole of run FM129, indicating rapid crystal growth. Magnetite varied in size (< 5-30 µm) and revealed near-perfect cubic crystal habit for most runs, sometimes accompanied by slightly rounded shapes. In run FM146, magnetite contained some melt and mineral inclusions pointing towards delayed nucleation followed by rapid crystallisation (Fig. 3f). Ilmenite was typically rather small (< 5-10 µm) and of elongated shape, occasionally forming crystal clusters. Orthopyroxene was only observed in a single experiment (FM164) and crystallised as relatively large idiomorphic prisms (up to 200 µm length) containing numerous inclusions. Weak sector zonation could be observed on BSE images, which was confirmed by compositional data established by SEM-EDS and EPMA.
Biotite crystallised in run FM145 (Fig. 3d) formed idiomorphic platelets (~ 20 µm), while apatite morphology differed between the two runs where it was present (short prismatic in FM145 and tiny needles in FM164).

Approach to equilibrium
The following considerations imply that chemical equilibrium was approached to a sufficient level in the polybaric fractional crystallisation experiments: (1) Blundy et al. (2020) showed that olivine Fe-Mg K D 's vary as a function of forsterite contents, with a maximum K D at Fo 50 . Consequently, the observed non-linear trend of decreasing Fe/Mg ratios is consistent with the decrease of olivine forsterite contents of our experiments with decreasing temperature (Fig. 8). Only the 990 °C run of the hot oxidised path falls off this general trend with a K D of 0.16, which is a result of the oxidising fO 2 conditions of this experiment (≈ RRO) and the significantly higher Fe 3+ content in the melt phase. Amphiboleliquid Fe-Mg distribution is illustrated in Fig. 4b. Except for the 930 °C run of the cold path (FM121), (Fe/Mg) ratios of amphiboles and residual melt display a linear trend with a fitted mean Fe-Mg K D of 0.334 ± 0.005. This distribution coefficient is in good accordance with the value of 0.32 from Grove et al. (2003), falls in the range of 0.30-0.38 established by Sisson and Grove (1993a), and agrees with values between 0.35 and 0.40 observed by Martel et al. (1999). The calculated K D of run FM121 (0.629 ± 0.037) is almost twice the fitted Fe-Mg K D of 0.334 ± 0.005. This run crystallised high proportions of magnetite, which could be an indication that fO 2 might have been significantly higher than expected (>> NNO), although no indications of fO 2 buffer failure were noticed. Enhanced incorporation of ferric iron in amphibole combined with abundant magnetite crystallisation could explain the distinctly higher Fe-Mg K D . Alternatively, the early saturation of amphibole and the delayed onset of magnetite crystallisation in the experiment circumvented a final equilibration in Fe-Mg exchange between amphibole and residual liquid. Thus, the (Fe/Mg) ratio of amphibole rather represents the equilibrium state prior to magnetite saturation, where the melt contained significantly higher amounts of iron and the remaining run duration after the onset of magnetite crystallisation was apparently too short to enable a final re-equilibration in Fe-Mg between already crystallised amphibole and residual melt. For clinopyroxene, a mean Fe-Mg K D of 0.276 ± 0.005 was established by weighted linear fitting of data points of experiments where fO 2 was buffered between NNO-1 and NNO. This value broadly coincides with 0.23 for a hydrous basalt (Sisson and Grove 1993a), 0.24 for a high-Mg andesite (Grove et al. 2003), and 0.25 for andesites (Martel et al. 1999). Clinopyroxenes from the hot oxidised path exhibit higher Fe-Mg K D 's ranging from 0.32 to 0.51 consistent with higher fO 2 (~ RRO) and an enhanced incorporation of ferric iron into clinopyroxene via ferri-tschermaks and acmite components.

Phase relations and phase proportions
Stable phase assemblages are illustrated in Fig. 5  The hot oxidised and hot rk52 path follow similar P-T ascent trajectories. However, due to differences in starting material composition and fO 2 , phase equilibria upon cooling and ascent differ (Fig. 5b). The first three experiments of the hot oxidised path (until 1080 °C and 0.8 GPa) are identical with the hot path. Upon decompression to 0.6 GPa and cooling to 1050 °C plagioclase and hercynitic spinel crystallise. At 0.4 GPa and 1020 °C, olivine disappears, but reappears at 0.2 GPa and 990 °C, where hercynitic spinel is replaced by magnetite. The initial starting material of the hot rk52 path is in equilibrium with olivine and clinopyroxene at 1170 °C and 1.0 GPa . This mineral assemblage is not changed by decompression to 0.8 GPa and a temperature decrease of 60 °C. The subsequent decrease to 0.6 GPa leads to the destabilisation of clinopyroxene at 1080 °C. Plagioclase joins the phase assemblage upon further isobaric cooling to 1050 °C. Finally, at 0.4 GPa and 1020 °C, the stable mineral assemblage (ol + cpx + plag) is identical to the previous fractionation experiment at 1050 °C. In summary, for several polybaric fractionation paths, decompression and cooling stabilised olivine at the expense of clinopyroxene, while plagioclase crystallised at higher temperatures at lower pressures, and amphibole, although present at intermediate pressures (0.4-0.6 GPa), is destabilised during final decompression to 0.2 GPa.
Phase proportions of experimental charges, calculated via mass balance regression, are reported in Table 2 and illustrated in Fig. 6. The left column (Fig. 6a, c, e, g, i) corresponds to instantaneous solid compositions (ISC) depicting the modal proportions of minerals crystallising in individual experiments. The right column (Fig. 6b, d, f, h, j) represents total solid compositions (TSC), where phase proportions are expressed relative to the initial starting material (rk54 or rk52), corresponding to the total accumulated amounts of minerals fractionated along the liquid line of descent. TSC fractions were calculated as follows: TSC n = TSC n-1 + ISC n * AMF n-1 , where n corresponds to the experiment of interest, n-1 to the preceding fractionation experiment, and AMF to the absolute residual melt fraction expressed relative to the initial starting material. For most experiments, instantaneous solid compositions are relatively constant and below 30 wt.%. A distinct decrease (more than 10 wt.%) is often related to a decompression step of 0.2 GPa, due to a shift of phase equilibria with decreasing pressure. This is further enhanced for water-undersaturated experiments because decompression decreases volatile solubility in silicate melts, and, thus, results in an increase of water activity in the experimental charge. The experiments at 0.2 GPa of the hot and hot oxidised path exhibit a strong increase in crystallinity. We propose two possible effects explaining this observation: (1) decompression leads to water-saturated conditions and crystallisation is mainly driven by cooling, and, (2) the negative Clapeyron slope of plagioclase and magnetite results in extensive crystallisation of these phases. In contrast, accumulated crystal fractions of the intermediate and cold path display continuously increasing mineral proportions with progressive cooling. A comparison of relative mineral proportions between the five different fractionation paths reveals that phase relationships between olivine, clinopyroxene, and plagioclase are crucial. For the coolingdominated ascent path (cold path) the extent of clinopyroxene crystallisation is higher with respect to the more decompression-controlled P-T trajectory (hot path). Relative proportions of crystallised plagioclase and olivine are slightly higher for the latter fractionation trend. For the hot oxidised path, higher amounts of oxides (hercynitic spinel or magnetite) formed due to distinctly more oxidising fO 2 conditions, stabilising Fe 3+ -bearing mineral phases. The hot rk52 path fractionated almost double the amount of crystals compared to the hot path (44 wt.% vs 24 wt.%) at 1020 °C. This is an expression of the more primitive character of the rk52 starting material promoting extensive crystallisation with ongoing cooling.

Compositions of major phases
Compositions of major experimental phases (residual melts and coexisting minerals) were measured by EPMA and are reported in Table 3 and the Electronic Supplementary Material ESM 6.

Residual liquids
The evolution of major and minor oxide compositions of residual liquids with decreasing temperature (on an anhydrous base) is illustrated in Fig. 7. Liquid lines of descent follow calc-alkaline differentiation trends with increasing silica and alkali contents accompanied by a decrease of other compounds (e.g. MgO, FeO, and CaO), thereby evolving from basaltic to dacitic compositions. However, some distinct features can be identified, which are either valid for all five fractional crystallisation paths or only affect individual trajectories. Since several fractionation series exhibit similar evolution trends, the following umbrella terms are used in the remainder of this manuscript to facilitate the description of the observed patterns: the hot and hot oxidised path are called decompression-dominated, while the intermediate and cold path are referred to as cooling-dominated.
Silica contents increase with decreasing temperature from 48 to 67 wt.% (Fig. 7a). In detail, this trend of silica enrichment can be divided in a high-temperature interval with only a minor increase in SiO 2 , followed by a low-temperature stage with pronounced silica enrichment over a narrow temperature range. The onset of SiO 2 enrichment is shifted to higher temperatures (990-1020 °C) for the decompression-dominated series compared to the cooling-dominated ones (960-990 °C) because of phase equilibria control. At high temperatures, the chemical evolution of residual liquids is governed by fractionation of olivine, clinopyroxene, and ± plagioclase. At lower temperatures, crystallisation of amphibole, anorthite-rich plagioclase, and an Fe-Ti-oxide results in a rapid increase of silica. Al 2 O 3 reveals distinct bell-shaped trends with an initial increase followed by a plateau and a subsequent decrease (Fig. 7c). For the decompression-dominated series, maximum levels of Al 2 O 3 are lower (< 20.0 wt.%) than for the cooling-dominated trajectories (> 20.0 wt.%), consistent with enhanced plagioclase stability at lower pressures. Similar to alumina, TiO 2 exhibits an initial enrichment followed by a rapid decrease with variable inflection points (i.e. temperatures) for the individual FC series (Fig. 7b). These inflexion temperatures can be related to the onset of amphibole crystallisation. With decreasing temperature, saturation in ilmenite or Ti-bearing magnetite further depletes the residual melt in titanium. The hot oxidised path does not exhibit a late-stage decrease in TiO 2 , although magnetite saturation was reached. This feature is explained by low TiO 2 (~ 2.9 wt.%) but high ferric iron contents in magnetite and the absence of amphibole in the crystallisation sequence.

Olivine
Olivine was stable over a wide temperature range (900-1110 °C) since pressure decrease considerably expanded the olivine stability field. Olivine re-appeared upon decompression to 0.

Clinopyroxene
Experimental clinopyroxenes range between diopside and augite and contain considerable amounts of Al 2 O 3 (2.1-8.5 wt.%, see Table 3). Sector zoning (e.g. in Al 2 O 3 and TiO 2 contents) was observed for two runs of the hot path (FM93 at 1050 °C and FM146 at 960 °C). There is no consensus if either one of the two sectors represents the "true" equilibrium composition to be expected at respective P-T conditions or if an average composition between both sectors would be more appropriate (e.g. Schwandt and Mckay 2006;Nandedkar et al. 2014). However, we infer that high-Al sectors represent equilibrium values. Clinopyroxene compositions derived by EPMA were normalised following the procedure of Wood and Banno (1973).
Tetrahedral aluminium contents Al IV (in atoms per formula unit, a.p.f.u.) exhibit a well-established negative linear correlation with the sum of ferrous iron and magnesium (Fe 2+ + Mg) (Fig. 9a). This trend is almost parallel to the ideal trajectory of the Tschermak's exchange vector (Fe 2+ ,Mg)Si = Al VI Al IV (black arrow in Fig. 9a) revealing that tetrahedral aluminium incorporation in experimental clinopyroxenes is strongly controlled via the Tschermak's substitution. Ti contents of clinopyroxenes (in a.p.f.u.) plotted against temperature show an initial increase followed by a late-stage depletion (Fig. 9b). These trends perfectly match TiO 2 evolution trajectories of coexisting residual liquids (Fig. 7b) indicating that Ti incorporation in clinopyroxene is controlled by the melt composition (and, thus, indirectly the saturation of Fe-Ti-oxides). Total aluminium (Al tot , in a.p.f.u.) as a function of temperature exhibits a bell-shaped behaviour (Fig. 9c). In an initial stage, Al in clinopyroxene increases with decreasing temperature, followed by a rapid decrease at lower temperatures. This trajectory is consistent with the Al 2 O 3 evolution of residual liquids, indicating that melt composition (and, therefore, plagioclase saturation) controls Al in clinopyroxene. In addition, increasing silica activity in the melt shifts the Tschermak's substitution to the silica-rich side resulting in a depletion of clinopyroxene in Al. Sodium contents of experimental clinopyroxenes exhibit a weak increase with ongoing cooling consistent with an enrichment of residual melts in Na 2 O (Fig. 9d)
Amphibole compositions were recalculated with the normalisation algorithm RAMP (Spear and Kimball 1984) with an empirical Fe 3+ /Fe tot ratio of 0.20, representing a value in the range established by Clowe et al. (1988) for a tschermakitic hornblende at fO 2 between QFM and NNO.
With decreasing temperature, Si contents of amphibole increase from 5.8 atoms per formula unit (a.p.f.u.) at 990 °C to 6.6 a.p.f.u. at 870 °C (Fig. 10a). Amphiboles synthesised in two low-temperature runs from the cold path (FM131, 900 °C and FM145, 870 °C) exhibit a faint chemical zonation (not visible in BSE images), most prominent in Ti concentrations. Si contents of low-Ti zones are higher compared to coexisting high-Ti ones (6.4 and 6.6 a.p.f.u. vs 6.2 and 6.3 a.p.f.u.) as Ti contents are controlled via the Ti-Tschermak's exchange (see below) where enhanced Ti incorporation  coincides with elevated Al and lower Si contents. Ti plotted against Al IV reveals well-defined positive correlations allowing the identification of two distinct subparallel trajectories (Fig. 10b). Slopes of Al IV vs Ti are parallel to the ideal Ti-Tschermak's vector ((Fe 2+ ,Mg)Si 2 = TiAl IV 2 , black arrow) implying that incorporation of Ti in amphiboles is controlled by the Ti-Tschermak's exchange reaction. The expression of two trajectories could be related to the rapid increase of Si contents between 930 and 900 °C resulting in lower Al IV and a change in co-crystallising Fe-Ti-oxides from Ti-bearing magnetite to ilmenite. Apparently, the coexistence with ilmenite slightly enhances the incorporation of Ti in amphiboles compared to the coexistence with magnetite (Ti content of 0.29 a.p.f.u. at 930 °C and 0.32 a.p.f.u. at 900 °C). Total alkali contents on the A-site (Na A + K) exhibit a positive correlation with tetrahedral aluminium and show a continuous decrease with decreasing temperature (Fig. 10c) consistent with the well-known temperature dependence of the edenite vector (▯Si = (Na,K)Al IV ) (e.g. Holland and Blundy 1994).
The attainment of experimental equilibrium was tested calculating crystallisation temperatures employing the richterite-edenite thermometer by Holland and Blundy (1994) and resulting temperatures deviate 1-41 °C from effective run temperatures, in excellent agreement within the reported algorithm uncertainty (35-40 °C). Temperatures established for low-Ti zones of zoned amphiboles are distinctively off from experimental conditions unlike the high-Ti zones indicating that high-Ti amphibole compositions represent "real" equilibrium, whereas low-Ti ones are experimental relics, probably corresponding to partly-reacted metastable clinopyroxene. The algorithms of Ridolfi et al. (2010) and Ridolfi and Renzulli (2012) resulted in significant deviations from run conditions and strongly overestimate experimental temperatures (up to 195 °C), with increasing offset with decreasing temperature. Similarly, derived pressures and melt H 2 O contents are highly variable and do not match experimental run conditions. Thus, the amphibole thermobarometers of Ridolfi et al. (2010) and Ridolfi and Renzulli (2012) failed to reproduce experimental run conditions.

Plagioclase
Plagioclase crystallised over a wide temperature range (1050-870 °C) and occurred at pressures between 0.2 and 0.8 GPa. Anorthite contents reveal that plagioclase is anorthite-rich (An 90 -An 77 ) and displays a consistent decrease with decreasing temperature for all five fractionation paths (Fig. 11a). The hot, hot oxidised, and intermediate path demonstrate a more pronounced decrease of anorthite contents for the last temperature step investigated, coinciding with an increase of coexisting residual liquids in Na 2 O (Fig. 7h). Although Na contents of the melt are also progressively  increasing along the cold path, the decrease in anorthite is less pronounced. This difference can be explained considering water-saturation levels of the various experimental series. The lowest temperature runs of the hot, hot oxidised and intermediate path were performed at 0.2 GPa and, thus, water-saturation was reached in these charges. For H 2 O-undersaturated magmas, crystallisation with decreasing temperatures enriches residual melts in H 2 O and, thus, increases water activity. H 2 O activity also increases due to decreasing H 2 O solubility in silicate liquids with decreasing pressure (e.g. Hamilton et al. 1964;Shishkina et al. 2010;Ghiorso and Gualda 2015). Sisson and Grove (1993a) demonstrated that increasing water activity leads to the stabilisation of anorthite relative to the albite component in plagioclase. Thus, increasing H 2 O activities in nominally water-undersaturated magmas with ongoing differentiation suppress a pronounced decrease of anorthite contents of plagioclase. Upon water-saturation, this effect vanishes, and plagioclase compositions are controlled by the Na-Ca systematics of coexisting liquids. Although the last two experiments (870 and 900 °C) of the cold path at 0.4 GPa reached water-saturation (Table 2), plagioclase anorthite contents do not show a distinct decrease and reasons for this behaviour are currently unclear. Ca-Na ratios between plagioclase and residual melt and K D trajectories are illustrated in Fig. 11b. It is evident that each explored fractionation path exhibits a strong dependence of Ca-Na distribution coefficients on experimental temperature. At high temperatures (≥ 1020 °C) K D 's scatter around 3.0-4.0 but evolve to higher values (4.0-5.5) with cooling finally reaching a range of 5.5 to 7.0 at lowest temperatures. Sisson and Grove (1993a) showed that water content (and/or water activity) exerts a strong influence on Ca-Na distribution coefficients reaching maximum Ca-Na K D values of about 5.5 at water-saturation at 0.2 GPa. These findings agree with our data documenting progressively increasing water contents (and water activities) upon differentiation.

Fe-Ti-oxides
Spinel group minerals exhibit a strong compositional dependence on temperature, where hercynitic spinel crystallised at high temperatures (1020-1050 °C), titanomagnetite formed at intermediate temperatures (930-960 °C), and Ti-bearing magnetite was stable at low temperatures (900-930 °C). This transition in spinel composition corresponds to the onset of ferric iron incorporation in the spinel structure at the expense of Al. Spinel group minerals of the hot oxidised path exhibit elevated ferric iron contents and diminished Ti concentrations compared to spinel from the other series because of the higher fO 2 of these experiments. Ilmenite crystallised along the cold path and coexists with magnetite at 960 °C and 0.6 GPa, but represents the only stable Fe-Ti-oxide in the two lowest temperature runs (900 and 870 °C). Generally, MgO contents of experimental ilmenite are minor (4.03-5.00 wt.%), and recalculated ferric iron contents range from 0.09 to 0.18 a.p.f.u.

Minor phases
Biotite was only stable in the lowest temperature run of the cold path (870 °C and 0.4 GPa) with an xMg of 0.70, an average TiO 2 content of 3.87 wt.%, and minor amounts of Na 2 O (1.10 wt.%). Similarly, orthopyroxene was only

Comparison of experimental liquid lines of descent with Cascades volcanic rocks
First, we compare the experimentally derived liquid lines of descent (LLD) with a compilation of bulk rock data from the Cascades volcanics in the Western U.S. (Du Bray et al. 2006) and with previous isobaric fractional and equilibrium crystallisation experiments. Since our polybaric fractionation series were terminated when residual liquids reached andesitic to dacitic compositions, more evolved (rhyolitic) compositions exhibited by the natural rocks were not reproduced. However, differentiation trends of intermediate and more evolved magmas are controlled by multicomponent phase equilibria and follow well-defined compositional trajectories as established by previous experimental studies (e.g. Grove and Baker 1984;Sisson and Grove 1993a;Blatter et al. 2017;Müntener and Ulmer 2018;Ulmer et al. 2018;Marxer and Ulmer 2019). It has to be emphasised that comparisons between experimental LLD's and natural rocks assume that the latter represent true liquid compositions, which is not entirely true for most cases and, thus, has to be kept in mind for the following discussion. Overall, LLD's of our polybaric fractionation series as well as previous experimental trends from the literature only partly overlap with the differentiation trends defined by the Cascades volcanic rocks. Due to the near-linear relationship with experimental temperature (Fig. 7f), we chose MgO to illustrate experimental and natural differentiation trends. Silica contents of the polybaric LLD's define a nonlinear increase with decreasing MgO similar to the Cascades rocks, with an excellent overlap at low MgO (≤ 3-4 wt.%) for the last fractionation steps of each polybaric series (Fig. 12a). At less evolved stages, residual melts plot at the lower end of the field delineated by natural rocks. One possible reason is the low SiO 2 of our starting material (SiO 2 of 47.4 wt.%). Primitive rocks from the Cascades volcanoes, however, are generally more enriched in silica. Furthermore, for any specific MgO content, LLD's defined by previous experimental studies plot at the lower end of natural rocks delineated by the Cascades, but generally above our own data. However, the differentiation trend of Nandedkar et al. 2014; reproduces well the natural spread, which could be related to the more silicic character of their starting material and the slightly lower initial water content favouring the attainment of a higher melt silica activity.
TiO 2 contents of experimental melts agree well with the natural rock compositions but essentially only reproduced the lower range of natural titanium contents (Fig. 12b). The elevated TiO 2 concentrations of some Cascades rocks can be explained by compositional variations of primary mantlederived magmas, where high TiO 2 rocks represent H 2 O-poor decompression melts (i.e. high-alumina olivine tholeiites) (e.g. Bartels et al. 1991;Grove et al. 2002), while rocks with lower TiO 2 contents correspond to subduction related magmas produced by fluid flux-melting of mantle rocks (e.g. Grove et al. 2002Grove et al. , 2005. Fractional crystallisation LLD's at 0.7-1.0 GPa Ulmer et al. 2018) also reproduce the natural differentiation trend albeit at lower maximum TiO 2 compared to polybaric fractionation. This can be related to the efficient depletion of residual melts in titanium due to enhanced amphibole fractionation at higher pressures and/or slightly higher fO 2 (NNO to NNO+1) favouring earlier saturation of Fe-Ti-oxides. The effect of amphibole fractionation on melt TiO 2 contents is also shown by the LLD at 0.4 GPa of Nandedkar (2014), where amphibole and Fe-Ti-oxide crystallisation was not observed resulting in an enrichment of the residual melts in TiO 2 . The LLD's by Blatter et al. (2013), albeit starting at higher TiO 2 contents, reveal a similar evolution, where the onset of TiO 2 decrease is almost pressure independent and agrees well with our polybaric data.
Experimental liquids are enriched in Al 2 O 3 with respect to the Cascades rocks (Fig. 12c). This is related to the combined effects of suppressed crystallisation of plagioclase at high pressures (0.6-0.8 GPa) and the elevated Al 2 O 3 contents of our primary starting materials (Table 1). This indicates that a majority of the Cascades volcanics did not fractionate exclusively in the lower crust, as rocks with comparable MgO concentrations reveal distinctly lower Al 2 O 3 contents than the experimental LLD's. Decompression of ascending magmas enhances plagioclase crystallisation, resulting in a decrease of Al 2 O 3 of experimental liquids following compositional trends of the natural rocks. In fact, only residual liquids of the last decompression steps to 0.2 GPa for the hot, hot oxidised, and intermediate path plot within the Cascades range, while the cold path remains at too high Al 2 O 3 contents due to pronounced fractionation at higher pressures. Isobaric differentiation trends between 0.7 and 1.0 GPa (Blatter et al. 2013;Nandedkar et al. 2014;Ulmer et al. 2018) closely follow our polybaric fractionation trends and likewise overestimate Al 2 O 3 contents with respect to the natural rocks. Only the 0.4 GPa LLD's reproduce the natural differentiation trend (Blatter et al. 2013;Nandedkar 2014) illustrating the effect of pressure on Al 2 O 3 contents of residual liquids. Our fractionation experiments overestimate iron contents ( Fig. 12d) with respect to the natural rock record at intermediate stages of differentiation (2-6 wt.% MgO). This could be related to delayed saturation of Fe-Ti-oxides as a consequence of slightly lower fO 2 of our experiments compared to the natural rock record. In contrast, the most evolved run of the hot oxidising path plots at too low FeO contents inferring that fO 2 conditions close to RRO put an upper bound on magmatic redox conditions. This interpretation is supported by experiments at more oxidising conditions, which better reproduce the natural rock record (NNO to NNO+1 for Nandedkar (2014), Nandedkar et al. (2014), andUlmer et al. (2018) or RRO-0.5 to RRO+1.3 for Blatter et al. (2013)).
CaO characteristics (Fig. 12e) of the polybaric fractionation paths are shifted towards higher values at intermediate (5-10 wt.%) MgO contents. Only upon the last fractionation steps, our data overlap with the natural rock record. This can be explained by higher CaO contents of our starting material compared to the Cascades volcanics. This interpretation is supported by previous experiments on related starting compositions from the Adamello which overlap with our data and also plot at too high CaO concentrations (Fig. 12e). However, LLD's established by Blatter et al. (2013) almost perfectly reproduce the Cascades trend indicating that CaO trajectories are dominantly controlled by the primary magma composition and less by the initial stages of differentiation.
K 2 O evolution trends closely reproduce the natural trend exhibited by the Cascades rocks (Fig. 12f), mostly covering the upper end of potassium concentrations. The last fractionation experiments of the hot, hot oxidised, and intermediate path plot at the upper boundary of the natural compositions. This is related to the incompatible behaviour of potassium in calc-alkaline systems and a slightly elevated K 2 O content of our primary starting material. Residual liquids of the equilibrium and fractional crystallisation experiments plot at slightly lower potassium contents but show a similar characteristic enrichment in K 2 O upon decreasing MgO (Fig. 12f). Figure 13a presents FeO tot /MgO ratios (all iron as FeO) as a function of SiO 2 content of residual melts to distinguish tholeiitic from calc-alkaline differentiation trends (Miyashiro 1974). Residual liquids from previous experimental series follow the boundary between the tholeiitic and calc-alkaline field during early stages of differentiation but enter the latter upon advanced crystallisation. Our data reveal a distinct excursion towards tholeiitic compositions with increasing SiO 2 followed by a back-bend into the calcalkaline field related to Fe-Ti-oxide crystallisation. On the contrary, the 0.7 GPa differentiation trends display only a weak initial enrichment in iron and become calc-alkaline between 55 and 60 wt.% of SiO 2 . The variations in FeO tot / MgO among different experimental series is controlled by fO 2 during differentiation as discussed in detail by Sisson et al. (2005). The combined effects of low fO 2 and low H 2 O contents result in delayed crystallisation of Fe-Ti-oxides, driving residual liquids into the tholeiitic field, while calcalkaline trajectories are favoured by more hydrous and oxidising conditions (e.g. Sisson et al. 2005;Blatter et al. 2013;Nandedkar et al. 2014).
In the TAS diagram (SiO 2 vs Na 2 O + K 2 O) the experimental series reproduce the natural differentiation trend (Fig. 13b). However, the high-pressure LLD's at 0.7 and 1.0 GPa evolve along the lower limit of the natural compositional field and reach rhyolitic compositions, while the polybaric trends rather plot at the upper end and only extend into the andesite or dacite field. These differences in TAS trajectories are related to variations in Na 2 O and K 2 O contents of the different starting materials (e.g. potassium) and/ or variable phase equilibria (e.g. stability fields of plagioclase and amphibole).
In summary, a detailed comparison of our experimental data and previous studies with the natural rock record from the Cascades reveals that polybaric fractional crystallisation only partly reproduces the chemical variability of arc rocks. Differences for some oxides (e.g. K 2 O or SiO 2 ) can be explained by variations in the experimental starting materials while others are related to the redox conditions during differentiation (e.g. FeO and FeO tot /MgO). Distinct differences among other components, such as Al 2 O 3 , are controlled by phase equilibria, i.e. the fractionation of clinopyroxene during early stages of differentiation at high pressures and the simultaneous suppression of plagioclase crystallisation. The compositional spread revealed by natural calc-alkaline rock suites requires additional magmatic processes besides polybaric fractionation, such as imperfect fractionation (i.e. crystal entrainment during melt extraction), magma mixing (e.g. Sisson et al. 2005;Reubi and Blundy 2009;Blatter et al. 2013), or crustal assimilation (e.g. DePaolo 1981; Hildreth and Moorbath 1988) and is addressed in the last section of the discussion.

Evolution of the alumina saturation index
Evolution trajectories of the alumina saturation index (ASI, molar ratio of Al 2 O 3 /(CaO + Na 2 O + K 2 O)) of experimental liquids are compared with volcanic rocks from the Cascades and the 1.0 GPa fractionation experiments of Ulmer et al. (2018) preceding the polybaric series (Fig. 14). Natural rocks follow a differentiation trend from metaluminous (ASI < 1) towards weakly peraluminous (ASI > 1) compositions. In contrast, our polybaric LLD's differ significantly and plot at elevated ASI's. Most remarkably, the coolingdominated fractionation trends (cold and intermediate path) rapidly evolve towards peraluminous compositions and become corundum-normative at silica contents between 48 and 52 wt.%. In contrast, the decompression-dominated fractionation trends (hot and hot oxidised path) remain metaluminous during differentiation and enter the compositional field covered by the natural rocks upon final decompression to 0.2 GPa. This difference between different polybaric fractionation series is directly related to the effect of pressure on phase equilibria. At high pressures (0.6-0.8 GPa), extensive clinopyroxene crystallisation results in a pronounced depletion of residual melts in CaO. Such an evolution towards andesitic or dacitic peraluminous differentiates at high pressures has previously been reported by experimental studies (e.g. Blatter et al. 2013;Nandedkar et al. 2014;Ulmer et al. 2018). Upon decompression to 0.4-0.2 GPa, clinopyroxene and amphibole destabilise at the expense of olivine and plagioclase keeping residual liquids metaluminous. Consequently, decompression-dominated fractionation trends better reproduce the natural compositional range.

Phase relations and liquid lines of descent in pseudoternary projections
Compositions of residual experimental liquids and the Cascades volcanics are plotted in the normative olivine (Ol) -clinopyroxene (Cpx) -quartz + orthoclase (Qtz + Or) pseudoternary diagram projected from plagioclase (Plag), apatite (Ap), and ilmenite-hematite (Ilm-Hem) to illustrate the control of phase equilibria on the LLD's (Fig. 15). Normative mineral compounds were re-calculated from melt compositions following the approach presented by Grove et al. (1992) and modified by Blatter et al. (2017). To facilitate the interpretation of LLD's, we additionally plotted bulk compositions of experimental cumulates (highlighted by the grey shaded background). Cumulate compositions were calculated based on phase proportions (Table 2) and average compositions of mineral phases (Table 3)  these rocks predominantly formed at moderately oxidising fO 2 conditions. All five LLD's exhibit a continuous decrease of normative clinopyroxene and olivine accompanied by an enrichment in quartz + orthoclase, illustrated by the differentiation trends away from the Ol-Cpx side-line towards the Qtz + Or corner. These trends are consistent with extensive crystallisation of Fe-Mg silicates with ongoing cooling (Figs. 7 and 12). Differentiation trajectories can roughly be distinguished in three intervals: an initial phase characterised by olivine fractionation resulting in an increase or stagnation of the clinopyroxene component, followed by a second interval dominated by clinopyroxene and olivine cotectic crystallisation depleting residual liquids efficiently in normative clinopyroxene, and a final interval, where the crystallisation of amphibole and magnetite leads to a pronounced enrichment of residual melts in SiO 2 shifting the trajectories towards the Qtz + Or corner. Such differentiation trends are a characteristic feature of calc-alkaline differentiation, as previously shown by numerous studies (Sisson and Grove 1993a;Pichavant and Macdonald 2007;Nandedkar et al. 2014;Ulmer et al. 2018). Experimental cumulates define two main groups. The first one lies close to the Ol apex and represents dunitic or troctolitic cumulates. The second group plots close to the Qtz + Or base-line at similar normative Ol and Cpx contents and either represents clinopyroxene-or amphibole-dominated cumulates. In detail, amphibole-bearing cumulates are shifted towards more negative Qtz + Or (due to nepheline-normative amphibole) while clinopyroxene-dominated ones are displaced to slightly higher Cpx.
During early stages of fractionation, melt compositions from the hot, intermediate, and cold path are similar, with an enhanced decrease of the clinopyroxene component accompanied by a less distinct reduction of the olivine compound. Subsequently, the hot path diverges from the more cooling-dominated intermediate and cold LLD's. The latter two trajectories continue their evolution towards the Cpx base-line, while the hot path bends towards the Qtz + Or corner as a consequence of enhanced olivine crystallisation and destabilisation of clinopyroxene, keeping residual melts metaluminous. Crystallisation of clinopyroxene and amphibole drives the cooling-dominated fractionation trends into the peraluminous compositional field. The intermediate path exhibits extensive orthopyroxene and magnetite fractionation in the last step directing residual liquids towards the Qtz + Or corner, while the LLD defined by the cold path is controlled by amphibole fractionation and evolves deeper into the peraluminous field. The hot rk52 path was explored to test if an elevated clinopyroxene component of the initial basaltic magma results in an LLD subparallel to the previous FC paths (hot, intermediate, or cold), but displaced to higher normative clinopyroxene. This was, however, not the case; it converges with the other series originating from the starting material rk54 upon the last temperature step investigated. Thus, starting crystallisation from a more Cpx-rich initial composition just results in higher proportions of early clinopyroxene fractionation but has no effect on the later stage chemical evolution of the residual liquids. In contrast, the hot oxidised path branches off from the hot path at 1080 °C and 0.8 GPa and evolves along a trajectory subparallel to the LLD's of the other fractionation series but maintains higher normative clinopyroxene components, underlining the important control of fO 2 on crystallisation-differentiation.
In principle, the exact location of the olivine-clinopyroxene cotectic curve controlling the LLD of arc magmas depends on pressure, fO 2 , and bulk system composition (i.e. xMg). The olivine-clinopyroxene cotectic is shifted towards clinopyroxene-rich compositions with decreasing pressure (i.e. polybaric fractionation) leading to a destabilisation of clinopyroxene at the expense of olivine (e.g. Grove et al. 1992;Hamada and Fujii 2008;Stamper et al. 2014;Melekhova et al. 2015) and a displacement of LLD's towards the Cpx apex. Consequently, cooling-dominated polybaric fractionation (cold path) exhibits an evolution towards strongly peraluminous compositions, while the LLD of decompression-dominated one (hot path) plots at higher normative clinopyroxene and remains in the metaluminous field. The intermediate path lies between these two endmembers, but also becomes weakly peraluminous upon progressive differentiation. An increase of fO 2 leads to the destabilisation of clinopyroxene at the expense of olivine. This effect is linked to the ferrous iron content of the crystallising magma. An increase of fO 2 results in reduced proportions of FeO in the melt at the expense of Fe 2 O 3 and an increase of the effective xMg (= Mg/(Mg + Fe 2+ )) of the residual melt favouring olivine over clinopyroxene (e.g. Ulmer et al. 2018). In fact, the hot oxidised path at RRO exemplifies this influence of fO 2 on the olivine-clinopyroxene cotectic, which is displaced towards higher normative Cpx component relative to NNO. In addition, elevated fO 2 conditions promoted crystallisation of spinel phases, further enhancing SiO 2 enrichment and FeO decrease of residual melts.
The comparison of the experimental LLD's with compositions of the Cascades volcanics reveals that polybaric differentiation cannot completely reproduce the natural rock record. The depletion of residual liquids in normative clinopyroxene contents along the cold and intermediate path drives them below the natural rock record into the peraluminous field. However, the intermediate path experiences a pronounced shift towards the quartz + orthoclase corner during the last differentiation step (fractionation of Fe-Ti-oxide bearing norites) resulting in an overlap with the natural rock record for weakly peraluminous compositions. The LLD of the hot path is slightly shifted towards higher normative clinopyroxene and follows the lower limit of the natural rocks overlapping in the last fractionation step with metaluminous intermediate to more evolved natural rocks. The hot oxidised path follows higher normative clinopyroxene contents overlapping with the Cascades volcanics.
Consequently, decompression-dominated magma ascent paths closely overlap with natural volcanic rocks, while cooling-dominated ascent trajectories deviate significantly. The present data emphasise the possible importance of fO 2 of silicate magmas on LLD's: only the hot oxidised path is able to reproduce the compositional trend of volcanic rocks inferring that redox conditions of fractionating arc magmas are, in fact, about one log-unit higher (NNO+1 to NNO+2) than commonly assumed (NNO to NNO+1) (e.g. Gill 1981;Cottrell et al. 2021).

General implications for arc magma differentiation
Our experiments show that decompression-dominated polybaric fractionation represents a possible scenario for arc magma differentiation, although the applicability of this model exhibits some limitations. Consequently, additional (open-system) magmatic processes and/or different near-adiabatic decompression paths with variable initial temperature need to be considered to derive a general model for arc magmatism able to explain the observed compositional variation of natural calc-alkaline rocks.
One process is mixing of melts with different compositions (i.e. basaltic and dacitic to rhyolitic) in the lower crust (e.g. Hildreth and Moorbath 1988). Thereby, the evolved magmas either represent differentiation products of basaltic magmas originating from the underlying mantle or lowdegree partial melts from previously solidified basaltic magmas re-melted by heat supply of freshly injected primitive melts. After mixing, these buoyant metaluminous intermediates start ascending towards the upper crust and further differentiate via polybaric fractionation. In fact, rocks from the Cascades exhibit broadly linear trajectories for several chemical parameters (Figs. 12,13,and 15) between compositions from evolved basalt (~ 8 wt.% MgO) to andesite/dacite (~ 2 wt.% MgO). However, primitive rocks do not match these linear trends pointing towards early crystallisationdifferentiation of primary mantle-derived basalts. Likewise, more evolved rhyodacitic to rhyolitic compositions do not represent extrapolations of the linear intermediate trajectories. This observation indicates that the late stage chemical evolution of residual liquids is, again, dominated by crystallisation processes. The distinct rheology contrast between mafic and felsic magmas is severely hampering any efficient mixing of strongly contrasting compositions and favouring magma mingling instead (e.g . Eichelberger 1975;Sparks and Marshall 1986;Blundy and Sparks 1992). However, in case of mixing components being close in composition and temperature (i.e. evolved basalt and andesite) this difference 3 Page 32 of 36 in rheology also diminishes potentially facilitating the mixing process.
A second, potentially even more important, mechanism is incomplete fractionation and crystal retention in decompressing magmas. Early fractionation of primitive basalt at the crust-mantle boundary is dominated by the crystallisation of olivine, clinopyroxene, and very minor Cr-rich spinel resulting in wherlite and olivine-clinopyroxenite cumulates. Derivative buoyant residual magmas rising towards the upper crust may contain significant proportions of this clinopyroxene-dominated crystal cargo. Upon ascent and decompression, the olivine-clinopyroxene cotectic shifts towards clinopyroxene by dissolving entrained clinopyroxene, while olivine remains stable and/or continues to crystallise/fractionate. Consequently, normative clinopyroxene contents of the ascending magma increase driving it towards the compositional range covered by natural rocks (Fig. 15).
In summary, we infer that the compositional variation revealed by natural calc-alkaline rocks can be reproduced by a combination of polybaric fractionation, mixing of magmas of differentiated basaltic and andesitic/dacitic composition, and the entrainment of clinopyroxene upon melt extraction in the lower crust. A conceptual model illustrating the various processes of arc magma differentiation is shown in Fig. 16. Hydrous mantle-derived primitive basalts fractionate olivine + clinopyroxene ± spinel, and minor amounts of plagioclase and amphibole in the lower crust. This fractionation process is accompanied by mixing between evolved basaltic and andesitic/dacitic magmas. Clinopyroxene and olivinedominated cumulate assemblages are potentially entrained due to imperfect fractionation processes in the lower crust. During decompression and magma ascent, residual liquids further evolve via polybaric fractionation of olivine, plagioclase, and magnetite and by dissolution or minor crystallisation of clinopyroxene. In addition, differentiation during ascent is potentially accompanied by mixing/mingling of compositionally similar magma batches in the conduit. Finally, resulting evolved magmas are either emplaced in the upper crust and solidify to intermediate/felsic plutonic rocks or erupt on the surface building arc volcanic complexes.

Conclusions
This experimental study explored the scenario of polybaric fractional crystallisation of arc magmas and tested its applicability to explain the compositional variation of natural calc-alkaline rocks at convergent plate margins. A total of five different magma ascent paths were explored to investigate the combined effects of cooling and decompression, fO 2 , and the composition of the initial starting material on the resultant liquid lines of descent. The most relevant findings of this study are: • During polybaric fractionation, decompression causes continuous changes in phase equilibria controlling the LLD's. With decreasing pressure, clinopyroxene is destabilised at the expense of olivine resulting in a shift of the olivine-clinopyroxene cotectic curve towards more clinopyroxene normative compositions. The stability field of plagioclase expands with decreasing pressure, with important effects on Al 2 O 3 contents of derivative liquids. Amphibole crystallisation was restricted to elevated pressures of 0.4-0.6 GPa, but absent in the lowest pressure runs at 0.2 GPa. • Liquid lines of descent established via polybaric fractional crystallisation follow calc-alkaline differentiation trends with continuously increasing silica and alkali contents and depletion in MgO. Differentiation trends of cooling-dominated fractionation evolve towards peraluminous melt compositions similar to previous high-pressure isobaric crystallisation studies and do not reproduce the compositional variation observed for calc-alkaline volcanic rocks from the Cascades (Western U.S.). For decompression-dominated fractionation, the pressure-induced shift of the olivine-clinopyroxene cotectic results in a slight deflection of LLD's towards higher normative clinopyroxene keeping them metaluminous, similar to the natural volcanic record. • Results from oxidised fractionation experiments demonstrate a fundamental control of fO 2 on the chemical evolution of calc-alkaline liquids. The LLD is shifted towards clinopyroxene-rich normative compositions resulting in an excellent match of experimental melts with natural bulk rock compositions. Consequently, our data indicate that fO 2 potentially plays a key role for arc magma differentiation opening new avenues for future studies on the quantification of varying redox conditions on calc-alkaline phase equilibria. • Based on the compositional variation revealed by typical calc-alkaline rocks, polybaric fractional crystallisation alone cannot explain the natural arc magmatic differentiation trend. Mixing between differentiated basalts and intermediate liquids (andesite and/or dacite) and crystal entrainment from previously formed clinopyroxene-dominated cumulates during melt segregation are important in modifying LLD's defined by fractional crystallisation. Both processes result in an enrichment of residual melts in normative clinopyroxene and, thus, circumvent the evolution of magmas towards peraluminous compositions. However, additional experimental and petrological research is required to test and refine this model.